New Delay Less Sub Band Adaptive Filtering Algorithm Foractive Noise

Global Journal of Researches in Engineering: f Electrical and Electronics Engineering Volume 14 Issue 1 Version 1.0 Year 2014 Type: Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4596 & Print ISSN: 0975-5861

New Delay Less Sub Band Adaptive Filtering Algorithm for Active Noise Control Systems By B. Satish Chandra, S. China Venkateswarlu, D. Ravi Kiran Babu

& K. Arun Kumar Jyothishmathi Institute of Technology & Science, India Abstract- The delay less SAF scheme in an ANC system involves the decomposition of input noise (i.e., the reference signal) and error signals into sub bands using analysis filter banks, and combining the sub band weights into a full-band noise canceling filter by a synthesis filter bank called weight stacking. Typically, a linear-phase finite-impulse response (FIR) low-pass filter (i.e., prototype filter) is designed and modulated for the design of such filter banks. The filter must be designed so that the side-lobe effect and spectral leakage are minimized. The delay in filter bank is reduced by prototype filter design and the side-lobe distortion is compensated for by oversampling and appropriate stacking of sub band weights. Experimental results show the improvement of performance and computational complexity of the proposed method in comparison to two commonly used sub band and block adaptive filtering algorithms.

Keywords: DFT, dolph-linear -phase finite-impulse response, sub band adaptive filtering, SAF, UDFM.

GJRE-F Classification : FOR Code: 091301, 090699

NewDelayLessSubBandAdaptiveFilteringAlgorithmforActiveNoiseControlSystems

Strictly as per the compliance and regulations of :

© 2014. B. Satish Chandra, S. China Venkateswarlu, D. Ravi Kiran Babu & K. Arun Kumar. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

New Delay Less Sub Band Adaptive Filtering Algorithm for Active Noise Control Systems

B. S atish Chandra α , S. China Venkateswarlu σ , D. Ravi Kiran Babu ρ & K. Arun Kumar Ѡ

Abstract- The delay less SAF scheme in an ANC system In general, the SAF methods offer a good involves the decomposition of input noise (i.e., the reference alternative approach to meet ANC system requirements, signal) and error signals into sub bands using analysis filter due to their inherent spectral decomposition and down banks, and combining the sub band weights into a full-band sampling operations. Since the spectral dynamic range noise canceling filter by a synthesis filter bank called weight and eigen value spread of the covariance matrix of noise 2014 stacking. Typically, a linear-phase finite-impulse response (FIR) low-pass filter (i.e., prototype filter) is designed and signal decrease in each sub band, the performance, i.e., Year modulated for the design of such filter banks. The filter must convergence rate, noise attenuation level, and stability be designed so that the side-lobe effect and spectral leakage of the ANC system, improves using SAF techniques.

35 are minimized. The delay in filter bank is reduced by prototype Hence, one expects that increasing the number of sub filter design and the side-lobe distortion is compensated for by bands (or block length) M should improve the oversampling and appropriate stacking of sub band weights. performance. Experimental results show the improvement of performance The delay less SAF scheme in an ANC system and computational complexity of the proposed method in involves the decomposition of input noise (i.e., the comparison to two commonly used sub band and block reference signal) and error signals into sub bands using adaptive filtering algorithms. Sub band adaptive filtering (SAF) techniques play a prominent role in designing active noise analysis filter banks, and combining the sub band control (ANC) systems. They reduce the computational weights into a full-band noise canceling filter by a complexity of ANC algorithms, particularly, when the acoustic synthesis filter bank called weight stacking. Typically, a noise is a broadband signal and the system models have long linear-phase finite-impulse response (FIR) low-pass filter impulse responses. In the commonly used uniform-discrete (i.e., prototype filter) is designed and modulated for the Fourier transform (DFT) -modulated (UDFTM) filter banks, design of such filter banks. The filter must be designed increasing the number of sub bands decreases the so that the side-lobe effect and spectral leakage are I I Version XIV Issue olume computational burden but can introduce excessive distortion, V

minimized. The latter requires a high-order FIR filter, ) degrading performance of the ANC system. In this paper, we F

introducing a long delay, which increases with M as the ( propose a new UDFTM-based adaptive sub band filtering method that alleviates the degrading effects of the delay and bandwidth shrinks. The long delay and side-lobe side-lobe distortion introduced by the prototype filter on the interference introduced by the prototype filter degrade system performance. the performance of SAF algorithms for large M, limiting

Keywords: DFT, dolph-linear-phase finite-impulse the computational saving that can be obtained by response, sub band adaptive filtering, SAF, UDFM. increasing the number of sub bands. Improving the system performance and reducing the computational I. Introduction burden by increasing M has inspired the work presented

ubband adaptive filtering (SAF) techniques play a herein. The focus of this project is to design a high- performance SAF algorithm. The performance limiting

prominent role in designing active noise control Researches in Engineering S(ANC) systems. They reduce the computational factors of existing SAF structures were found to be due complexity of ANC algorithms, particularly, when the to the inherent delay and side-lobes of the prototype filter in the analysis filter banks. acoustic noise is a broadband signal and the system models have long impulse responses. Active noise A delay less structure targeted for low-resource of ournal control (ANC) is a method of canceling a noise signal in implementation is proposed to eliminate filter bank an acoustic cavity by generating an appropriate anti- processing delays in sub band adaptive filters obal J

Gl (SAFs). Rather than using direct IFFT or poly phase noise signal via canceling loudspeakers. filter banks to transform the SAFs back into the

time-domain, the proposed method utilizes a weighted Author α: Assoc. Prof -ECE JITS Jyothishmathi Institute of Technology & overlap-add (WOLA) synthesis. Low-resource real-time Science, Nustulapur, Karimnagar -505 481, Andhra Pradesh, India. Author σ: P rof. & HOD -ECE JITS Jyothishmathi Institute of Technology implementations are targeted and as such do not & Science, Nustulapur, Karimnagar-505 481, Andhra Pradesh, India. involve long (as long as the echo plant) FFT or IFFT e-mail: [email protected] operations. Also, the proposed approach facilitates time Author ρ: Assoc. Prof -ECE JITS Jyothishmathi Institute of Technology &

Science, Nustulapur, Karimnagar-505 481, Andhra Pradesh, India. distribution of the adaptive filter reconstruction Author Ѡ: Asst. Prof-ECE SCITES Jyothishmathi Institute of Technology calculations crucial for efficient real-time and hardware & Science, Nustulapur, Karimnagar-505 481, Andhra Pradesh, India. implementation. The method is implemented on an

oversampled WOLA filter bank employed as part of an matrix. Specifically when N − P ≥ M, the N ×P echo cancellation application. Evaluation results polynomial (resp. : Laurent polynomial) of M-variety demonstrate that the proposed implementation matrix is generically invertible; whereas when N − P < outperforms conventional SAF systems since the signals M, the matrix is generically noninvertible. Using this used in actual adaptive filtering are not distorted by filter sharp phase transition property, we develop a fast bank aliasing. The method is a good match for partial algorithm to compute a particular left inverse for a given update adaptive algorithms since segments of the time- Laurent polynomial matrix. These results suggest an domain adaptive filter are sequentially reconstructed alternative approach in designing multidimensional filter and updated. banks by freely generating filters for the analysis side A delay less method for adaptive filtering first. If we allow an amount of over sampling then we can through SAF systems is proposed. The method, based almost surely find a perfect reconstruction inverse for on WOLA synthesis of the SAFs, is very efficient and is the synthesis poly phase matrix. These results also have well mapped to a low-resource hardware potential applications in multidimensional signal

2014 implementation. The performance of an open-loop reconstruction from multi-channel filtering and sampling. version of the system was compared against a Speech signals from the uncontrolled environments may Year conventional SAF system employing the same WOLA contain degradation components along with the analysis/synthesis filter banks, with the proposed delay required speech components. The degradation

36 less system offering superior performance but at greater components include background noise, reverberation computational cost. The performance is identical to the and speech from other speakers. The degraded speech DFT-FIR delay less SAF system that employs gives poor performance in automatic speech straightforward poly phase filter banks. processing tasks like speech recognition and speaker However, the proposed WOLA -based TAF recognition and is also uncomfortable for human synthesis offers a superior mapping to low-resource listening [1]. The degraded speech therefore needs to hardware with limited-precision arithmetic. Also the be processed for the enhancement of speech WOLA adaptive filter reconstruction may easily be components. Several methods have been proposed in spread out in time simplifying the necessary hardware. the literature for this purpose, majority them can be This time-spreading may be easily combined with partial grouped into spectral processing and temporal

e XIV Issue I Version I I Version XIV Issue e update adaptive algorithms to reduce the computation processing methods. In spectral processing methods,

um cost for low-resource real time platforms. the degraded speech is processed in the transform domain, where as, in temporal processing methods, the Vol II. Generic Invertibility of processing is done in the time domain, for enhancing

() Multidimensional Fir Multirate the speech components. Each of them has their own merits and demerits. These two approaches may be Systems and Filter Banks effectively combined by exploiting their merits and We study the inevitability of M-variety aiming to minimize the demerits. This may lead to polynomial (respectively: Laurent polynomial) matrices speech enhancement methods which are more effective of size N by P. Such matrices represent and robust compared to only spectral or temporal multidimensional systems in various settings including processing. filter banks, multiple-input multiple-output systems, and Frequency -domain and sub band implementa- MultiMate systems. The main result of this paper is to tions improve the computational efficiency and the prove that when N − P ≥ M, then H(z) is generically convergence rate of adaptive schemes. The well-known Researches in Engineering invertible; whereas when N − P < M, then H(z) is multi delay adaptive filter (MDF) belongs to this class of generically noninvertible. As a result, we can have an block adaptive structures and is a DFT-based algorithm. alternative approach in design of the multidimensional In this paper, we develop adaptive structures that are

ournal of ournal systems. During the last two decades, one dimensional based on the trigonometric transforms DCT and DST multiage systems in digital signal processing were and on the discrete Hartley transform (DHT). As a result,

obal J thoroughly developed. In recent years, due to the high these structures involve only real arithmetic and are Gl demand in multidimensional processing including image attractive alternatives in cases where the traditional DFT- and video processing, volumetric data analysis and based scheme exhibits poor performance. The filters are spectroscopic imaging, multidimensional multirate derived by first presenting a derivation for the classical systems have been studied more extensively. DFT-based filter that allows us to pursue these

One key property of a multidimensional extensions immediately. The approach used in this

multirate system is its perfect reconstruction, which paper also provides further insights into sub band guarantees that an original input can be perfectly adaptive filtering. reconstructed from the outputs. We show that there is a sharp phase transition on the invariability depending on the size and dimension of a given Laurent polynomial

III. The Implementation of Delay Less of operation requires a cyclic pattern of collaboration Sub Band Active Noise Control among the nodes, and has the advantage that for the last node in the cycle, the data from the entire network Algorithms are used to update the desired parameter estimate, Wideband active noise control systems usually thereby offering excellent estimation performance. have hundreds of taps for control filters and the Moreover, for every measurement, every node cancellation path models, which results in high needs to communicate with only one neighbor. computational complexity and low convergence speed. However, incremental cooperation has the disadvantage Several active noise control algorithms based on sub of requiring the deﬁnition of a cycle, and network band adaptive filtering have been developed to reduce processing has to be faster than the measurement the computational complexity and to increase the process, since a full communication cycle is needed for convergence speed. The sub band structure is similar to every measurement. This may become prohibitive for the frequency domain structure but differs in the time large networks. Incremental networks are also less domain processing of the sub band signals. This paper robust to node and link failures. An alternative protocol 2014 discusses several issues associated with implementing is the diffusion implementation where every node the delay less sub band active noise control algorithms communicates with all of its neighbors as dictated by Year on a DSP Platform, such as the modeling of the the network topology. This approach has no topology

37 cancellation path in sub bands and the partial Update of constraints and is more robust to node and link failure. It different sub bands. will have some performance degradation compared to Single channel ANC systems often use sub an incremental solution, and also every node will need band techniques to overcome the difficulties of high to communicate with its neighbors for every computational complexity and low convergence speed measurement, possibly requiring more energy than the associated with a wideband control filter containing incremental case. thousands of taps. The mainstay of the proposed model is This paper will discuss various method of noise improving the system performance and reducing the reduction for wireless communication network. Noise is computational burden. In this paper, we first an, unwanted and inevitable interference, in any form of demonstrate that the increased delay degrades the communication. It is non-informative and plays the role system performance more than that of the spectral of sucking the intelligence of the original signal. Any kind leakage (or side-lobe effects) in a uniform sub-band filtering method. It is shown how the spectral leakage I I Version XIV Issue olume of processing of the signal contributes to the noise V addition. A signal traveling through the channel also can be reduced by choosing a proper decimation factor ) F gathers lots of noise. It degrades the quality of the and weight stacking methodology. We then present a ( information signal. The effect of noise could be reduced new SAF (Sub-Band Adaptive Filtering) algorithm that only at the cost of the bandwidth of the channel, which reduces computational complexity by increasing the is again undesired, as bandwidth is a precious resource. number of subbands M without degrading the Hence to regenerate original signal, it is tried to reduce performance of the ANC (Active Noise Control) system. the power of the noise signal, or in the other way, raise The performance of the proposed method is compared the power level of the Informative signal, at the receiver with those of MT (Moragan and Thi) and DFT-MDF end this leads to improvement in the signal to noise (Discrete Fourier Transform and Multi-Delay Adaptive ratio(SNR). Filter) methods. The results show that the maximum Adaptive algorithms that allow neighboring noise attenuation level (NAL) of the proposed method is Researches in Engineering nodes to communicate with each other at every higher than that of MT and comparable to that of the iteration. At each node, estimates exchanged with DFT-MDF method. However, the new method achieves neighboring nodes are fused and promptly fed into the the maximum NAL with much lower computational local adaptation rules. In this way, an adaptive network complexity and higher robustness than the other two of ournal is obtained where the structure as a whole is able to methods.

obal J respond in real-time to the temporal and spatial IV. ethodology

M Gl variations in the statistical proﬁle of the data. Different adaptation or learning rules at the nodes, allied with The gradient based adaptation starts with an different cooperation protocols, give rise to adaptive old optimization technique known as the method of networks of various complexities and potential. steepest descent. This has been discussed in the next Obviously, the effectiveness of any distributed chapter in detail. It is recursive in the sense that starting implementation depends on the modes of cooperation from some initial arbitrary value for tap weight vector, it that are allowed among the nodes. Figure 1 illustrates improves with increasing number of iterations. The final three such modes of cooperation. In an incremental value so computed for tap weight vector converges to mode of cooperation information ﬂows in sequential Wiener solution. The fixed step size least mean square manner from one node to the adjacent node. This mode (FSS LMS) algorithm is an important member of the

©2014 Global Journals Inc. (US) New Delay ess Sub Band Adaptive Filtering Algorithm for Active Noise Control Systems family of stochastic gradient algorithms. The term 1 stochastic gradient is intended to distinguish it from the µ(n) = method of steepest descent that uses deterministic 2 ( ) ( ) gradient in a recursive computation of the Wiener filter This μ(n) is then substituted𝑇𝑇 into the standard for stochastic inputs. This algorithm does not require LMS recursion replacing 𝑥𝑥μ, resulting𝑛𝑛 𝑥𝑥 𝑛𝑛 in the following measurements of the pertinent correlation functions, nor NLMS equation. does it require matrix inversion. Subsequent works have discussed issue of optimization of step size or methods w(n+1) = w(n) + 2μ(n)e(n)x(n) , of varying step size to improve performance. There are 1 w(n+1)=w(n)+ ( ) ( ) different types of adaptive filtering algorithms, they are ( ) ( ) 1. Least mean square (LMS) algorithm. 2. Normalized least mean square (NLMS) algorithm. 3. Variable step c) Implementation of the𝑇𝑇 NLMS algorithm𝑒𝑒 𝑛𝑛 𝑥𝑥 𝑛𝑛 size LMS (VSLMS) algorithm. 4. Variable step size The NLMS algorithm 𝑥𝑥 𝑛𝑛 𝑥𝑥has𝑛𝑛 been implemented in Matlab and in a real time application using the Texas 2014 Normalized LMS (VSNLMS) algorithm. 5. Recursive least Instruments TMS320C6711 Development Kit. As the squares (RLS) algorithm.

Year step size parameter is chosen based on the current

a) Normalized least mean Square (NLMS) Algorithm input values, the NLMS algorithm shows far greater 38 One of the primary disadvantages of the LMS stability with unknown signals. This combined with good algorithm is having a fixed step size parameter for every convergence speed and relative computational

iteration. This requires an understanding of the statistics simplicity makes the NLMS algorithm ideal for the real of the input signal prior to commencing the adaptive time adaptive echo cancellation system. As the NLMS is filtering operation. In practice this is rarely achievable. an extension of the standard LMS algorithm, the NLMS Even if we assume the only signal to be input to the algorithms practical implementation is very similar to adaptive echo cancellation system is speech, there are that of the LMS algorithm. Each iteration of the NLMS still many factors such as signal input power and algorithm requires these steps in the following order amplitude which will affect its performance. (a) The output of the adaptive filter is calculated. The normalized least mean square algorithm 1 (NLMS) is an extension of the LMS algorithm which y(n) = ( ) = ( ) ( ) e XIV Issue I Version I I Version XIV Issue e =0 bypasses this issue by selecting a different step size um (b) An error signal𝑁𝑁− is calculated as the difference𝑇𝑇 between value, μ(n), for each iteration of the algorithm. This step ∑𝑖𝑖 � � Vol size is proportional to the inverse of the total expected the desired signal 𝑤𝑤and𝑛𝑛 the𝑥𝑥 filter𝑛𝑛 − output𝑖𝑖 𝑤𝑤 𝑛𝑛 𝑥𝑥 𝑛𝑛

energy of the instantaneous values of the coefficients of F

e(n)=d(n)-y(n) () the input vector x(n). This sum of the expected energies of the input samples is also equivalent to the dot (c) The step size value for the input vector is calculated. product of the input vector with itself, and the trace of 1 input vectors auto-correlation matrix, R. µ(n) = ( ) ( ) 1 2 2 ( ) = [ ( )] =0 (d) The filter tap weights are𝑇𝑇 updated in preparation for 𝑁𝑁− the next iteration. 𝑥𝑥 𝑛𝑛 𝑥𝑥 𝑛𝑛 𝑖𝑖1 2 𝑡𝑡𝑡𝑡 =𝑹𝑹 [ ∑ 𝐸𝐸( 𝑥𝑥 𝑛𝑛)] − 𝑖𝑖 =0 w(n +1) = w(n) +μ(n)e(n)x(n)

Researches in Engineering The recursion formula𝑁𝑁− for the NLMS algorithm is 𝑖𝑖 Each iteration of the NLMS algorithm requires stated in equation.𝐸𝐸 ∑ 𝑥𝑥 𝑛𝑛 − 𝑖𝑖 3N+1 multiplications, this is only N more than the standard LMS algorithm, this is an acceptable increase 1 ournal of ournal w(n+1)=w(n)+ ( ) ( ) considering the gains in stability and echo attenuation ( ) ( ) achieved.

obal J b) Derivation of the NLMS𝑇𝑇 algorithm𝑒𝑒 𝑛𝑛 𝑥𝑥 𝑛𝑛 Gl To derive the 𝑥𝑥NLMS𝑛𝑛 𝑥𝑥 algorithm𝑛𝑛 consider the V. Active Noise Control System standard LMS recursion, for which we select a variable A ctive noise control (ANC) is a method of step size parameter, μ(n). This parameter is selected so canceling a noise signal in an acoustic cavity by + that the error value, e (n), will be minimized using the generating an appropriate anti-noise signal via updated filter tap weights, w(n+1), and the current input canceling loudspeakers. Due to recent advances in + vector, x(n). w(n+1) = w(n) + 2μ(n)e(n)x(n), e (n) = d(n) wireless technology, new applications of ANC have T T – w (n+1)x(n) , =(1– 2μ(n)x (n)x(n))e(n) emerged, e.g., incorporating ANC in cell phones, Next we minimize (e+(n))2, with respect to μ(n). Bluetooth headphones, and MP3 players, to mitigate the Using this we can then find a value for μ (n) which forces environmental acoustic noise and therefore improve the e+(n) to zero. speech and music quality. For practical purposes, ANC

© 2014 Global Journals Inc. (US) New Delay ess Sub Band Adaptive Filtering Algorithm for Active Noise Control Systems as a real-time adaptive signal processing method noise) in amplitude and frequency, but has opposite should meet the following requirements: 1) minimum phase. Combination of these signals results in computational complexity (lower computational delay cancellation of the primary (unwanted) noise. This ANC and power consumption), 2) stability and robustness to technique is well-known for its use in cancelling input noise dynamics, and 3) maximum noise unwanted sound as shown by Nelson & Elliott [6], but it attenuation. is used for the control of vibration. A block diagram of Acoustical noise can sometimes disturb or even an adaptive digital filter is shown in fig.2.1, where n is a harm nearby people. Hence, it is necessary to find ways time index. This filter forms the basis for feedforward to reduce such unwanted noise. Traditionally, passive ANC, based on the FXLMS algorithm. The adaptive filter means (i.e., physical barriers) to attenuate the noises actually consists of two parts. The digital filter, W(z) have been employed. Unfortunately, the barriers are not calculates its output by using a reference x(n) and effective to isolate lower frequency noises; and to adjustable filter coefficients, or weights. The filter achieve significant reduction the barriers have to be coefficients are updated by an adaptive algorithm, using rather bulky. In effect, the passive barrier is not a cost- x(n) and an error signal e(n) in such a way that the 2014 effective solution to reducing low-frequency noises (for squared error e2(n) is minimized. example, noises that come from industrial blowers, Year diesel engines, transformers, earth-moving machines,

39 and propeller-driven aircraft.) Because of that shortcoming of the physical barriers, active means to reduce low frequency noise (less than 500-1000 Hertz) have been investigated by researchers in the field of adaptive acoustic control. Active noise control (ANC) promises a good reduction of the noises in the form of a small package of a DSP controller, microphone(s), and loudspeaker(s). For the better or the worse, the ANC systems are effective only when the intended noise is periodic, and so random noises like the white noise will not be reduced. There are different ANC schemes that have Figure 5.1 : ANC using an adaptive filter been developed. My project is involved with the I I Version XIV Issue olume We can define the error e(n) as: V

implementation of one of the schemes that is called )

e(n) = d(n) -y(n) (5.1) F

single-channel adaptive feedback ANC. The ( implementation was on a Texas Instruments where d(n) is an unwanted disturbance. The TMS320C54 evaluation module (EVM) board; in addition adaptive filter will try to calculate an output y(n) that is to this, I used a microphone and a loudspeaker. Two equal to the unwanted disturbance d(n), so this types of noise exist in the environment, broadband disturbance will be cancelled. noise, where its energy is more or less evenly distributed across the frequency spectrum, or narrowband noise, a) Concept of an ANC system where the energy is mostly concentrated around specific The basic concept of the feedforward ANC frequencies. In ANC roughly two types of control system that is used with the experimental setup can be strategies can be distinguished as shown by Fuller, their found in Figure 2.2, where the grey part represents the use strongly depends on the deterministic behavior of controller and the white part represents the physical Researches in Engineering the disturbance: world. This is a very general concept, in this report Feedback ANC: A controller is used to vibrations are considered, but it can also be applied to acoustic applications as shown by Nelson and Elliott [6] modificate the response of a system, for example by of ournal adding artificial damping. In this way vibration levels can or more specific to sound cancellation in ducts as shown by Kuo and Morgan [5]. be reduced even for a broadband random disturbance. obal J

Feedforward ANC: When the disturbance is b) System Description Gl deterministic, or in particular harmonic, a controller can The harmonic noise is produced at the noise be used to adaptively calculate a signal that cancels the source (e.g. an engine or a shaker). disturbance. When vibrations are induced by rotating Through the transfer function P(z) of the primary machinery this often results in harmonicvibrations and path this results in a vibration d(n) somewhere in the the amount of noise reduction achieved by feedforward construction. This vibration will be reduced, by ANC systems is far superior to that of feedback ANC generating the appropriate controller output y(n) and systems as shown by Hansen & Snyder. The basic idea sending it trough the transfer function S(z) of the of feedforward ANC is to generate a signal (secondary Secondary Path to the construction. The remaining noise), that is equal to a disturbance signal (primary vibration e(n) can then be measured by a sensor. The

adaptive filter looks similar to that of Figure 2.1 but is generally used. This algorithm places an estimate of slightly more complicated. That is to compensate for the S(z) in the reference signal to the weight update. effects of the Secondary Path, which will be explained For the ANC system of Figure 2.2, containing a later. Secondary Path transfer function S(z), the residual error can be expressed as: c) Conventional versus Indirect Feedforward ANC In conventional feedforward ANC systems, the e(n)=d(n)–y`(n); (5.2)

disturbance frequency information is available or can be where y`(n) is the output of the Secondary Path derived from the noise source, for example from the S(z). If S(z) is assumed as an IIR filter with denominator engine velocity. When the disturbance frequency is coefficients [a ,………, a ] and numerator coefficients exactly known, the reduction that can be achieved by a 1 N [b ,…….b ], then the filter output y`(n) can be written conventional feedforward ANC system has its limit at 0 M-1 as the sum of the filter input y(n) and the past filter infinity for the ideal case with a pure harmonic noise-free output: disturbance and linear Secondary Path. In other 2014 applications the disturbance frequency information may

Year not be available, because the disturbance frequencies (5.3)

are unknown or slowly varying. In that case indirect

40 feedforward ANC can be used as shown in this report,

where the reference signal x(n) is generated from the error e(n), instead of from the frequency information of It can be achieved in a similar way that the the noise source. Conventional feedforward ANC with a gradient estimate becomes: single frequency disturbance was implemented on the experimental setup by H.J. van der Veen. This report (5.4) focuses on different kinds of indirect feedforward ANC methods, where if possible harmonic disturbances with where: two frequencies are used. They are tested at the experimental setup and will be compared with each (5.5) other. e XIV Issue I Version I I Version XIV Issue e um Note that in practical applications, S(z) is not Vol exactly known, therefore the parameters ai and bj are the F

parameters of the Secondary Path Estimate S^ (z). The () weight update equation of the FXLMS algorithm is:

w(n + 1) = w(n) + µx`(n)e(n) (5.6)

and x`(n) can be calculated from Equation 5.5. The FXLMS algorithm is very tolerant to modelling errors in the Secondary Path Estimate S^ (z) as shown by Kuo & Morgan [5]. The algorithm will converge when the phase error between S(z) and S^

Researches in Engineering (z) is smaller than 900. Convergence will be slowed down though, when the phase error increases. From the weight update Equation 2.6 can be seen that a step size µ has to be chosen. This step size affects important ournal of ournal Figure 5.1 : Block diagram of an ANC system properties such as performance, stability and error after convergence. A more in-depth analysis can be found in obal J In practical applications there is a transfer

Gl function S(z) between the digital controller signal and Kuo & Morgan [5] and Elliott & Nelson. Furthermore, a modification of the standard FXLMS is presented to the physical world, which contains the D/A converter, power amplifier, actuator element and construction. In make the choice of µ independent of the power of x`(n).

general, this Secondary Path transfer function S(z) gives VI. System Analysis a change in amplitude and a phase shift, so the adaptive filter should compensate for the effects of S(z) a) Adaptive Filter Theory to ensure convergence. A straightforward solution would In the past three decades, interest in adaptive be to place the inverse S(z)-1 in series with S(z), but systems has increased, leading to widespread use of because this inverse does not necessarily exists, the so- adaptive techniques in fields such as Communications, called Filtered-x LMS (FXLMS) algorithm is more Signal Processing, Sonar and Biomedical Engineering.

Adaptive systems adapt to the environment changes and search for the optimal system parameters based on Input x(n) Desired a reference signal. In the case of a filter, the system Output y(n) Response d(n) parameters are the tap weights of the filter. The Linear Discrete time performance of an adaptive algorithm is highly Filtering + dependent on the reference input and additive noise statistics. In the context of Wiener filter theory, there are assumptions of time invariance, linearity and Gaussian statistics such that the mean square error criteria will be the optimum cost function. These assumptions are often for the ease of mathematical analysis, but do not take Estimation into account of the broader problems of signals with Error e(n) non-Gaussian statistics. In the digital communication systems, efficient bandwidth utilization is economically Figure 6.1 : Prototype Wiener Filtering Scheme 2014 important to maximizing profits, while at the same time The requirement is to make the estimation error Year maintaining performance and reliability. More as small as possible in some Statistical sense by importantly, the adaptive filter solution has to be

controlling impulse response coefficients w0, w1, ……., 41 relatively simple, which often leads to the use of the wN-1 . Two basic restrictions are: 1. The filter is linear, conventional Least Mean Square (LMS) algorithm. which makes mathematical analysis easy to handle.2. However, the performance of the LMS algorithm is often The filter is an FIR (symmetrical and odd ordered) filter. sub-optimal and the convergence rate is small. This, The filter output is y(n) and the estimation error therefore, provides the motivation to explore and study is given by e(n). The performance of the filter is variable step size LMS adaptive algorithms for various determined by the size of the estimation error, that is, a applications. smaller estimation error indicates a better filter b) The Wiener Filter performance. As the estimation error approaches zero, These are a class of linear optimum discrete the filter output y(n) approaches the desired signal d(n). time filters known collectively as Wiener filters. Wiener Clearly, the estimation error is required to be as small as filters are a special class of transversal Finite impulse possible. In simple words, in the design of the filter parameters, an appropriate function of this estimation response (FIR) filters that build upon the Mean Square I I Version XIV Issue olume

error as performance or cost function is chosen and the V Error (MSE) cost function to arrive at an optimal filter tap set of filter parameters is selected, which optimizes the ) weight vector, which reduces the MSE signal to a F minimum. Theory for a Wiener filter is formulated for cost function. In Wiener filters, the cost function is ( general case of complex valued time series with filter chosen to be specified in terms of its impulse response because x = E[e(n)2] (6.1) baseband signal appears in complex form under many practical situations. Where E[.] denotes the expectation or ensemble average since both d(n) and x(n) are random c) Mean Square Error Criterion processes. The linear filter with the aim of estimating the desired signal d(n) from input x(n). Assume that d(n) d) Wiener Filter: Transversal, Real valued case and x(n) are samples of infinite length, random Consider an adaptive transversal filter as shown Researches in Engineering processes illustrates in Fig 3.1 . In ‘optimum filter in Fig 3.2. Assume that the filter input x(n) and the design’, signal and noise are viewed as stochastic desired response d(n) are real valued stationary processes. The filter is based on minimization of the processes. The filter tap weights w0, w1,…………wN-1 are mean square value of the difference between the actual also assumed to be real valued , where N equals the of ournal filter output and some desired output, as shown in number of delay units or tap weights. fig.3.1. The filter input x(n) and tap weight vectors, w, obal J

can be defined as column vectors, Gl

x(n) = [ x(n) x(n-1)…… x(n-N+1)] (6.2) T w = [w0 w1 ……… wN-1]

The filter output is defined as

( ) = = w x(n-i) =wTx(n) = xT(n)w(n) (6.3) =0 i S ubsequently, 𝑛𝑛the error signal can be written as 𝑦𝑦 𝑛𝑛 ∑𝑖𝑖

e(n) = d(n)-y(n) = d(n) – wTx(n) = d(n) – xT(n)w (6.4) E[(e(n)2] = E[(d(n)- wT x(n)) (d(n)-xT (n)w)] ((6.6) Substituting (3.5) into (3.1) , the cost function is Expanding the last expression of (6.6) we obtained as, obtain, E[ d(n)2] – E[ d(n)xT(n)w] – E[d(n) wT x(n)]+ E[wT x(n) xT (n)w] (6.7)

Since w is not a random variable, E[d(n)2] –E[ d(n)xT (n)]w –wT E[d(n) x(n)]+wT E[ x(n) xT (n)]w (6.8)

x(n) x(n-1) x(n-N+1) Next, we can express E[d(n) x(n)] as an N x 1

-1 -1 cross correlation vector z-1 z z T p=E[d(n)x(n)]=[p0, p1, ……………pN-1] (6.9)

2014 T wn w0 w1 And E[x(n) x (n)] as a N x N autocorrelation matrix R T Year R=E[x(n)x (n)]= d(n) y(n)

42 00 01 02 0, 1 + + + (6.10) 𝑁𝑁− 𝑟𝑟 1𝑟𝑟,0 𝑟𝑟1,1 ⋯ 𝑟𝑟 1, 1 Tap Weight Control � ⋮ ⋱ ⋮ � Mechanism T T T Fr𝑟𝑟𝑁𝑁−om (6.9),𝑟𝑟𝑁𝑁− p = E[d⋯(n) x (n)𝑟𝑟𝑁𝑁−] and𝑁𝑁 hence− p w = wT p This implies that E[d(n) xT(n)]w = E[d(n) x(n)]wT.

Figure 6.1 : Structure of an Adaptive Transversal Filter Subsequently, we get

=E [d(n)2] – E[ d(n)xT(n)]w – wTE[d(n) x(n)]+ wTE[ x(n) xT(n)]w, =E[d(n)2]–2pT w + wT R w (6.11)

e XIV Issue I Version I I Version XIV Issue e This is a quadratic function of tap weight vector Where w0 indicates the optimum tap weight

um ‘w’ with a single global minimum. To obtain the set of vector. This equation is known as the Wiener Hopf filter tap weights that minimizes the cost function, , equation and can be solved to obtain the tap weight Vol solve the system of equations that results from setting vector, which corresponds to the minimum point of the F

() the partial derivatives of with respect to every tap cost function.

weight of the filter i.e. the gradient vector to zero. That is e) Iterative Search Algorithm It has been shown in the previous section that the Wiener Hopf equation can be solved to obtain the (6.12) = 0 optimum filter tap weights by minimizing a cost function, 𝜕𝜕𝜕𝜕 if the required statistics of the underlying signals ‘R’ and ‘p’ are available. Although this method is For i = 0, 1, ………..N𝜕𝜕𝑤𝑤𝑖𝑖 -1 where N = number of tap weights The gradient vector in (3.12) can also be straightforward, it presents serious computational difficulties, especially when the filter contains a large expressed as =0 (6.13) number of tap weights and the input data rate is high. Researches in Engineering Where Ñ is the gradient operator defined as column vector An alternative is to use an iterative search algorithm that starts at some arbitrary initial point in the tap weight vector space and moves progressively towards the

ournal of ournal = (6.14) optimum filter tap weight vector in steps. Each step is 𝜕𝜕𝜕𝜕1 𝜕𝜕𝜕𝜕2 𝜕𝜕𝜕𝜕 1 chosen with the aim of reducing the cost function. The obal J and 0 on the right hand side of (3.13) denotes the principle of finding the optimum filter tap weight vector Gl 𝛻𝛻 �𝜕𝜕𝑤𝑤 𝜕𝜕𝑤𝑤 𝜕𝜕𝑤𝑤𝑁𝑁 − � column vector consisting of N zero. It has been further by progressive minimization of the underlying cost function by means of an iterative algorithm is central to proved that the partial derivatives of x with respect to the filter tap weights can be solved such that the development of adaptive algorithms (e.g. LMS). In simplified terms, adaptive algorithms are actually  = 2Rw–2p (6.15) iterative search algorithms derived for minimizing the cost function by replacing the true statistics with By letting  = 0, the following equation is estimates obtained. obtained, in which the optimum set of Wiener filter tap weights can be obtained, Rw = p This implies that f) Meth o d of Steepest Descent Assume that the cost function to be minimized -1 w = R p = w0 (6.16) is convex (If the cost function corresponds to a convex

© 2014 Global Journals Inc. (US) New Delay ess Sub Band Adaptive Filtering Algorithm for Active Noise Control Systems quadratic surface, it has a unique minimum point. In b) Use this initial guess to compute the gradient vector other words, when the cost function is convex, the of the cost function with respect to the tap weights iterative search algorithm is guaranteed to converge to at the present point. the optimum solution), we may start with an arbitrary c) Update the tap weights by taking step in the point on the performance surface and take a small step opposite direction (sign change) of the gradient in the direction in which the cost function decreases vector obtained in step 2. This corresponds to step fastest. This corresponds to a step along the steepest in the direction of the steepest descent in the cost descent slope of the performance at that point. function at the present input. Furthermore, the size Repeating this successively, convergence towards the of the step is chosen proportional to the size of the bottom of the performance surface (corresponding to gradient vector. the set of parameters that minimize the cost function) is d) Go back to Step 2, and iterate the process until no guaranteed. further significant change is observed in the tap The method of steepest descent is an alternate weights i.e. the search has converged to an optimal 2014 iterative search method to find w0 (in contrast to solving point. the Wiener Hopf equation directly). The method of According to the above procedures, if w(n) is Year steepest descent algorithm belongs to a family of the tap weight vector at the nth iteration, then the iterative methods of optimization. It is a general scheme following recursive equation may be used to update 43 that performs an iterative search for a minimum point of w(n). any convex function of a set of parameters. Here, this ( + 1) = ( ) (6.17) method is implemented in transversal filter with the convex function referring to the cost function and the set Where µ is the positive scalar called step size, 𝑛𝑛 of parameters referring to the filter tap weights. It uses and nξ de𝑤𝑤notes𝑛𝑛 the gradient𝑤𝑤 𝑛𝑛vector− µ∆evaluated𝜉𝜉 at the the following procedures to search the minimum point of point w = w(n). the cost function of a set of filter tap weights. g) E∆rror Performance Surface a) Begin with an initial guess of the filter tap weights Th e estimation error e(n) can be given as: whose optimum values are to be found for

minimizing the cost function. Unless some prior 1 ( ) = ( ) =0 ( ) (6.18) knowledge is available, the search can be initiated by setting all the filter tap weights to zero, i.e. w(0). The cost function can be written𝑁𝑁− as

𝑖𝑖 𝑖𝑖 I I Version XIV Issue olume

𝐸𝐸 𝑛𝑛 𝑑𝑑 𝑛𝑛 − ∑ 𝑤𝑤 𝑥𝑥 𝑛𝑛 − 𝑖𝑖 V 2 1 = [ ( ) ] [ ( ) ( )] (6.19) )

=0 F

( 𝑁𝑁− ∗ ∗ The cost function or the mean squared𝑖𝑖 error is𝑖𝑖 computed for tap weight vector converges to Wiener precisely a second order𝜉𝜉 function𝐸𝐸 𝑑𝑑 of𝑛𝑛 the tap− ∑weights𝑤𝑤 in 𝐸𝐸solution.𝑥𝑥 𝑛𝑛 − 𝑖𝑖 𝑑𝑑 𝑛𝑛 the filter. Since ‘w’ can assume a continuum of values in The LMS algorithm has been extensively the N dimensional w-plane, the dependence of the cost analyzed in literature and a large number of results on function depends on the tap weights w0, w1, ……...wN-1 its steady state misadjustment and tracking may be visualized as a bowl shaped (N+1)-dimensional performance have been obtained. The fixed step size surface with N degrees of freedom represented by the least mean square (FSS LMS) algorithm is an important tap weights of the filter. The surface so described is member of the family of stochastic gradient algorithms. called the error performance surface of the transversal The term ‘stochastic gradient’ is intended to distinguish Researches in Engineering filter. The surface is characterized by a unique minimum, it from the method of steepest descent that uses where the cost function attains its minimum value. At deterministic gradient in a recursive computation of the this point, the gradient vector ∆ is identically zero. The Wiener filter for stochastic inputs. This algorithm does height corresponds to the physical description of not require measurements of the pertinent correlation of ournal filtering the signal x(n-i) with theξ fixed filter weight w, functions, nor does it require matrix inversion. from which a prediction error signal e(n) with power of Subsequent works have discussed issue of optimization obal J

Gl of step size or methods of varying step size to improve is generated. Some filter setting w0=(wo0,wo1) will performance. produce t he minimum MSE (wo is the optimum filter tap weight vector). This theory is the base of basic adaptive h) Performance of an Adaptive Algorithm algorithms of adaptive signal processing. The gradient The factors that determine the performance of based adaptation starts with an old optimization an algorithm are clearly stated below. Essentially, the te of technique known as the method of steepest descent. It most important factors as described here 1. Ra is recursive in the sense that starting from some initial Convergence: This is defined as the number of iterations arbitrary value for tap weight vector, it improves with required for the algorithm to converge to its steady state increasing number of iterations. The final value so mean square error. The steady state MSE is also known

as the Mean asymptotic square error or MASE. 2. x(n) x(n-1) x(n-N+1) Misadjustment: This quantity describes steady-state z-1 z-1 z-1 behavior of the algorithm. This is a quantitative measure of the amount by which the ensemble averaged final

w1 value of the mean-squared error exceeds the minimum w0 mean-squared error produced by the optimal Wiener wN-1 filter. The smaller the misadjustment, the better the x x x asymptotic performance of the algorithm. 3. Numerical Robustness: Th e implementation of adaptive filtering algorithms on a digital computer, which inevitably operates using finite word-lengths, results in quantization errors. These errors sometimes can cause + numerical instability of the adaptation algorithm. An 2014 adaptive filtering algorithm is said to be numerically y(n) robust when its digital implementation using finite -word- - Year length operations is stable. 4. Computational Requirements: This is an important parameter from a + 44 e(n) d(n) practical point of view. The parameters of interest include the number of operations required for one Figure 6.2 : Adaptive filter block diagram complete iteration of the algorithm and the amount of When the adaptive filter output is equal to memory needed to store the required data and also the desired signal the error signal goes to zero. In this program. These quantities influence the price of the situation the echoed signal would be completely

computer needed to implement the adaptive filter. cancelled and the far user would not hear any of their

5. Stability: An algorithm is said to be stable if original speech returned to them. the mean-squared error converges to a final (finite) value. Ideally, one would like to have a computationally VII. Least Mean Square (LMS) simple and numerically robust adaptive filter with high Algorithm e XIV Issue I Version I I Version XIV Issue e rate of convergence and small misadjustment that can

um be implemented easily on a computer. In the The LMS algorithm is a type of adaptive filter

Vol applications of digital signal processing e.g. adaptive known as stochastic gradient-based algorithms as it echo cancellation, the above factors play an important utilizes the gradient vector of the filter tap weights to

() role. converge on the optimal wiener solution. It is well known and widely used due to its computational simplicity. With There are different types of adaptive filtering algorithms, they are each iteration of the LMS algorithm, the filter tap weights

• Least mean square (LMS) algorithm of the adaptive filter are updated according to the following formula • Normalized least mean square (NLMS) algorithm w(n +1) = w(n) + 2μe(n)x(n) (7.1) • Variable step size LMS (VSLMS) algorithm • Variable step size Normalized LMS (VSNLMS) Here x(n) is the input vector of time delayed input values algorithm x(n) = [x(n) x(n-1) x(n-2) ….. x(n-N+1)]T (7.2)

Researches in Engineering • Recursive least squares (RLS) algorithm. The vector w(n) represents the coefficients of

i) The structure of Adaptive filter the adaptive FIR filter tap weight vector at time n. T The block diagram for the adaptive filter method w(n) = [w0(n) w1(n) w2(n) ….. wN-1(n)] (7.3) ournal of ournal utilized in this section. Here w represents the coefficients The parameter μ is known as the step size of the FIR filter tap weight vector, x(n) is the input vector -1 obal J samples, z is a delay of one sample periods, y(n) is the parameter and is a small positive constant. This step

Gl size parameter controls the influence of the updating adaptive filter output, d(n) is the desired echoed signal and e(n) is the estimation error at time n. The aim of an factor. Selection of a suitable value for μ is imperative to the performance of the LMS algorithm, if the value is too adaptive filter is to calculate the difference between the desired signal and the adaptive filter output, e(n). This small the time the adaptive filter takes to converge on error signal is fed back into the adaptive filter and its the optimal solution will be too long; if μ is too large the adaptive filter becomes unstable and its output coefficients are changed algorithmically in order to minimize a function of this difference, known as the cost diverges. function. In the case of acoustic echo cancellation, the a) Derivation of the LMS algorithm optimal output of the adaptive filter is equal in value to The derivation of the LMS algorithm builds upon the unwanted echoed signal. the theory of the wiener solution for the optimal filter tap

iii. The tap weights of the FIR vector are updated in weights, wo, as outlined in section 3.2.2. It also depends on the steepest descent algorithm as stated in equation preparation for the next iteration, by equation 7.9

3.23, this is a formula which updates the filter w(n +1) = w(n) + 2μe(n)x(n) (7.10) coefficients using the current tap weight vector and the current gradient of the cost function with respect to the The main reason for the LMS algorithms filter tap weight coefficient vector, ξ(n) popularity in adaptive filtering is its computational

simplicity, making it easier to implement than all other w(n+1)=w(n)-µ ξ(n) commonly used adaptive algorithms. For each iteration Where ξ (n)=E[e(n)2] (7.4) the LMS algorithm requires 2N additions and 2N+1 multiplications (N for calculating the output, y(n), one for As the negative gradient vector points in the 2 e(n) and an additional N for the scalar by vector direction of steepest descent for the N-dimensional μ multiplication). One of the primary disadvantages of the quadratic cost function, each recursion shifts the value LMS algorithm is having a fixed step size parameter for of the filter coefficients closer toward their optimum every iteration. This requires an understanding of the 2014 value, which corresponds to the minimum achievable statistics of the input signal prior to commencing the

value of the cost function, ξ(n). Year adaptive filtering operation. In practice this is rarely The LMS algorithm is a random process achievable. Even if we assume the only signal to be implementation of the steepest descent algorithm, from 45 input to the adaptive echo cancellation system is equation 3.23. Here the expectation for the error signal speech, there are still many factors such as signal input is not known so the instantaneous value is used as an power and amplitude which will affect its performance. estimate. The steepest descent algorithm then becomes The normalized least mean square algorithm (NLMS) is equation 3.24. an extension of the LMS algorithm which bypasses this w(n+1)=w(n)-µ ξ(n) issue by selecting a different step size value, μ(n), for 2 (7.5) each iteration of the algorithm. This step size is Where ξ (n) = e(n) proportional to the inverse of the total expected energy The gradient of the cost function, ▼ξ(n), can of the instantaneous values of the coefficients of the alternatively be expressed in the following form. input vector x(n). This sum of the expected energies of ξ(n)=(e2(n)) the input samples is also equivalent to the dot product 2 of the input vector with itself, and the trace of input ( ) vectors auto-correlation matrix, R. I I Version XIV Issue olume = V )

1 2 F

( ) = [ ( ] ) ( 𝜕𝜕𝑒𝑒 𝑛𝑛 =0 ( ) ( ( ) ( )) 𝑁𝑁− (7.11) = 2e(n) = 2e(n)𝜕𝜕𝜕𝜕 =(7.6) 𝑖𝑖 1 2 𝑡𝑡𝑡𝑡 𝑹𝑹= [ ∑=0 𝐸𝐸( 𝑥𝑥 𝑛𝑛 ) ] − 𝑖𝑖 𝜕𝜕𝜕𝜕 𝑛𝑛 𝜕𝜕 𝑑𝑑 𝑛𝑛 −𝑦𝑦 𝑛𝑛 𝑁𝑁− ( ) ( ) The recursion𝐸𝐸 ∑𝑖𝑖 formula𝑥𝑥 𝑛𝑛 for the − 𝑖𝑖 NLMS algorithm is 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 stated in equation 3.31. 2e(n) = 2e(n)x(n) 𝑇𝑇 1 Substitu𝜕𝜕𝑤𝑤ting this𝑛𝑛 into𝑥𝑥 𝑛𝑛the steepest descent w(n+1)=w(n)+ ( ) ( ) (7.12) ( ) ( ) algorithm of equation 3.8, we arrive at the recursion for 𝜕𝜕𝜕𝜕 Researches in Engineering the LMS adaptive algorithm. 𝑇𝑇 c) Derivation of the NLMS𝑥𝑥 𝑛𝑛 algorithm𝑥𝑥 𝑛𝑛 𝑒𝑒 𝑛𝑛 𝑥𝑥 𝑛𝑛 w(n +1) = w(n) + 2μe(n)x(n) (7.7) To derive the NLMS algorithm we consider the standard LMS recursion, for which we select a variable b) Implementation of the LMS algorithm of ournal step size parameter, μ (n). This parameter is selected so Each iteration of the LMS algorithm requires 3 that the error value, e+(n), will be minimized using the distinct steps in this order: updated filter tap weights, w(n+1), and the current input obal J Gl + i. The output of the FIR filter, y(n) is calculated using vector, x(n). w(n+1) = w(n) + 2µ(n)e(n)x(n) , e (n) = d(n) T T equation 3.27 – w (n+1)x(n), =(1– 2µ(n)x (n)x(n))e(n) (7.13) 1 Next we minimize (e+(n))2, with respect to μ(n). y(n)= =0 ( ) ( )= ( ) ( ) (7. 8) Using this we can then find a value for μ(n) which forces 𝑁𝑁− 𝑇𝑇 e+(n) to ii. The value𝑖𝑖 of the error estimation is calculated using ∑ 𝑤𝑤 𝑛𝑛 𝑥𝑥 𝑛𝑛 − 𝑖𝑖 𝑤𝑤 𝑛𝑛 𝑥𝑥 𝑛𝑛 equation 3.28. 1 zero.µ(n)= (7.14) e(n)=d(n)-y(n) (7.9) 2 ( ) ( ) 𝑇𝑇 𝑥𝑥 𝑛𝑛 𝑥𝑥 𝑛𝑛©2014 Global Journals Inc. (US) New Delay ess Sub Band Adaptive Filtering Algorithm for Active Noise Control Systems

This μ(n) is then substituted into the standard This requires prior knowledge of the input signal which

LMS recursion replacing μ, resulting in the following is not feasible for the echo cancellation system. NLMS equation. Algorithm: NLMS Algorithm w(n+1) = w(n) + 2µ(n)e(n)x(n) Average attenuation: -27.9dB

1 Multiplication operations: 3N+1 w(n+1)=w(n)+ ( ) ( ) (7.15) ( ) ( ) Comments: Simple to implement and computationally efficient. Shows very good attenuation and variable step 𝑇𝑇 size allows stable performance with non-stationary d) Implementation of the NLMS algorithm𝑒𝑒 𝑛𝑛 𝑥𝑥 𝑛𝑛 The NLMS algorithm𝑥𝑥 𝑛𝑛 𝑥𝑥 has𝑛𝑛 been implemented in signals. This was the obvious choice for real time

Matlab and in a real time application using the Texas implementation. Instruments TMS320C6711 Development Kit. As the Algorithm: VSSLMS Algorithm step size parameter is chosen based on the current Average attenuation: -9.8 dB 2014 input values, the NLMS algorithm shows far greater Multiplication operations: 4N+1 stability with unknown signals. This combined with good Year convergence speed and relative computational Comments: Displays very poor performance, possibly due to non-stationary nature of speech signals. Only half

46 simplicity makes the NLMS algorithm ideal for the real time adaptive echo cancellation system. the attenuation of the standard LMS algorithm. Not As the NLMS is an extension of the standard considered for real time implementation.

LMS algorithm, the NLMS algorithms practical Algorithm: VSSNLMS Algorithm

implementation is very similar to that of the LMS Average attenuation: -9.9 dB algorithm. Each iteration of the NLMS algorithm requires Multiplication operations: 5N+1 these steps in the following order. Comments: Increase in multiplications gives negligible The output of the adaptive filter is calculated. improvement in performance over VSSLMS algorithm. 1 The real time acoustic echo cancellation system y(n)= =0 ( ) ( ) = was successfully developed with the NLMS algorithm.

e XIV Issue I Version I I Version XIV Issue e 𝑁𝑁− The system is capable of cancelling echo with time

um 𝑖𝑖 ( ) ( ) (3.35) delays of up to 75 ms, corresponding to reverberation ∑ 𝑤𝑤 𝑛𝑛 𝑥𝑥 𝑛𝑛 − 𝑖𝑖

Vol off an object a maximum of 12 meters away. This proves An error signal𝑇𝑇 is calculated as the difference quite satisfactory in emulating a medium to large size

F between the desired𝑤𝑤 signal𝑛𝑛 𝑥𝑥 and𝑛𝑛 the filter output

() room.

e(n)=d(n)-y(n) (7.16) The utility of SBC is perhaps best illustrated with a specific example. When used for audio compression,

The step size value for the input vector is calculated. SBC exploits what might be considered a deficiency of the human auditory system. Human ears are normally 1 µ(n) = (7.17) sensitive to a wide range of frequencies, but when a 2 ( ) ( ) sufficiently loud signal is present at one frequency, the ear will not hear weaker signals at nearby frequencies. The filter tap weights𝑇𝑇 are updated in preparation 𝑥𝑥 𝑛𝑛 𝑥𝑥 𝑛𝑛 We say that the louder signal masks the softer ones. for the next iteration. The louder signal is called the masker, and the point at Researches in Engineering w(n +1) = w(n ) +μ(n)e(n)x(n) (7.18) which masking occurs is known, appropriately enough, as the masking threshold. The basic idea of SBC is to Each iteration of the NLMS algorithm requires enable a data reduction by discarding information about ournal of ournal 3N+1 multiplications, this is only N more than the frequencies which are masked. The result differs from standard LMS algorithm, this is an acceptable increase the original signal, but if the discarded information is

obal J considering the gains in stability and echo attenuation chosen carefully, the difference will not be noticeable, or Gl achieved. more importantly, objectionable. VIII. Comparison of Adaptive Filtering IX. Encoding Audio Signals Algorithms The simplest way to digitally encode audio

Algorithm: LMS Algorithm signals is pulse-code modulation (PCM), which is used on audio CDs, DAT recordings, and so on. Digitization Average attenuation: -18.2 dB transforms continuous signals into discrete ones by Multiplication operations: 2N+1 sampling a signal's amplitude at uniform intervals and Comments: Is the simplest to implement and is stable rounding to the nearest value representable with the

when the step size parameter is selected appropriately. available number of bits. This process is fundamentally

© 2014 Global Journals Inc. (US) New Delay ess Sub Band Adaptive Filtering Algorithm for Active Noise Control Systems inexact, and involves two errors: discretization error, the analysis and synthesis window lengths were from sampling at intervals, and quantization error, from increased to La 1024 and Ls 256 samples to better rounding. observe the effects of the delay. To simplify the analysis, The more bits used represent each sample, the batch synthesis was used for TAF WOLA reconstruction. finer the granularity in the digital representation, and This leads to a filter reconstruction delay of La / 2 512 thus the smaller the error. Such quantization errors may samples. Delaying the input signals by the same be thought of as a type of noise, because they are amount so that they are synchronized with the plant effectively the difference between the original source could compensate for the filter reconstruction delay. Of and its binary representation. With PCM, the only way to course this is counter productive as it creates delays in mitigate the audible effects of these errors is to use an otherwise delayless system. The ERLE drops at 30 enough bits to ensure that the noise is low enough to be seconds, and stays low for around 64 msecs masked either by the signal itself or by other sources of (corresponding to 512 samples of delay) before it starts noise. A high quality signal is possible, but at the cost of to rise again. This low-time of ERLE causes a drop in a high bitrate (e.g., over 700 kbit/s for one channel of echo cancellation performance and creates artifacts in 2014 CD audio). In effect, many bits are wasted in encoding the output. Repeating the experiment with delay masked portions of the signal because PCM makes no compensation, the ERLE drops later and start to rise Year assumptions about how the human ear hears. More right away as shown in the figure. The echo plant swap

47 clever ways of digitizing an audio signal can reduce that is unlikely to happen in practice; rather gradual plant waste by exploiting known characteristics of the auditory variations might occur. system. A classic method is nonlinear PCM, such as mu-law encoding (named after a perceptual curve in X. Simulation Results auditory perception research). Small signals are The input file ‘file1.wav’ is read by the command digitized with finer granularity than are large ones; the waveread and impulse response of secondary path is effect is to add noise that is proportional to the signal plotted as shown in fig 5.1. This means the output of strength. Sun's Au file format for sound is a popular secondary path y`(n) initially. Then estimating the example of mu-law encoding. Using 8-bit mu-law secondary propagation path S^(z) and identifying this encoding would cut the per-channel bit rate of CD audio path using NLMS adaptive filter. This is shown in fig 5.2 down to about 350 kbit/s, or about half the standard and also shows the required signal d(n),output signal rate. Because this simple method only minimally exploits y`(n) and error signal e(n).If the number of iterations are masking effects, it produces results that are often I I Version XIV Issue olume increased, the error signal e(n) is reduced. Here d(n) V 4 audibly poorer than the original. Sub-band coding is and y`(n) are having same signal value from 0.5*10 to ) F

4 used for example in G.722 codec. It uses sub-band 3*10 . ( adaptive differential pulse code modulation (SB- ADPCM) within a bit rate of 64 kbit/s. In the SB-ADPCM technique used, the frequency band is split into two sub-bands (higher and lower) and the signals in each sub-band are encoded using ADPCM. As explained in Section 2, in the proposed algorithm the TAF is obtained with a delay relative to the input signal. The amount of delay depends on the method of filter reconstruction. For sequential synthesis Researches in Engineering the delay is (La Ls) / 2 samples while for batch synthesis it is La / 2 samples. All of the delayless SAF methods reviewed in Section 1 have to deal with a plant reconstruction delay. The delay leads to a of ournal “synchronization problem” between the input signals and the plant, causing problems in tracking a dynamic obal J plant. The extent of the problem depends on the plant Gl time-dynamics and the TAF reconstruction delay. To demonstrate the effects of the delay, we simulated the system with the same system set up and input signals as described in the previous section with the following changes. The echo plant was switched to a new plant after 30 seconds through the experiment. With the employed analysis/synthesis filters used in the experiments, tracking problems were barely observable due to the low reconstruction delay of the system. Thus, Figure 8.1 : Secondary path filter response

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Figure 8.4 : Primary path filter response Figure 8.2 : Secondary path identification Fig 8.4 shows the primary propagation path a) Using NLMS algorithm filter response P(z).The output of P(z) is d(n),this is the Fig 8.3 shows accuracy of the estimated actual noise to be canceled by generating anti-noise secondary propagation path S^(z). Also the summer through the S^(z).Then the noise in the system is to be output e(n) in time-domain. Here the d(n) and y`(n) canceled and fig 5.5 shows the power spectral density e XIV Issue I Version I I Version XIV Issue e follows the same path and error signal e(n) is zero. of the canceled noise d(n). Power spectral density um means the distribution of d(n) over frequency-axis. The Vol voice frequency range is considered as from (0.3-3.5)

F KHz. () Fig 8.6 shows residual error signal spectrum of

e(n) or power spectral density of d(n) and y`(n).From (0- 1300)Hz, there is small difference between d(n) and y`(n).After that both are equal. Fig 5.7 shows the power spectral density of d(n)-y`(n).At 200Hz, power/frequency is -50 db/Hz (0.0031) approximately zero. The total complexity is plotted in Fig.5.8, number of real multiplications versus the number of subbands M. The plot is for the PFFT-2 method with

Researches in Engineering LSAF = 4N/M, as it results in better performance than that of PFFT-1. For comparison purposes, included the computational complexity of the MT and DFT -MDF algorithms. As shown, the computational complexities of ournal of ournal all methods reduce almost exponentially with M. The proposed technique compared to the other methods for obal J

Gl small values of M has higher computational complexity.

Figure 8.3 : Accuracy of Secondary path

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Figure 8.5 : Power spectral density of canceled Figure 8.7 : Comparison of computational complexity per input sample versus number of subbands (M)

XI. Conclusion

Acoustic paths such as those encountered in

ANC application usually have long impulse responses, which require longer adaptive filters for noise cancellation. Subband adaptive filters working with a large number of subbands have been shown to be a good solution to this problem. The focus of this project olume XIV Issue I Version I I Version XIV Issue olume

was to design such a high-performance SAF algorithm. V

The performance limiting factors of existing SAF ) F

structures were found to be due to the inherent delay ( and side-lobes of the prototype filter in the analysis filter

banks. Hence, the analysis filter banks were modified to reduce the inherent delay. A new weight stacking transform was designed to alleviate the interference introduced by the side-lobes. The modifications resulted in a new subband method that, unlike existing methods, improves the performance and reduces the computational complexity for a large number of

Figure 8.6 : Residual error signal spectrum Noise subbands. Researches in Engineering Experimental results showed that the proposed The new technique works very well with a larger method outperformed the two commonly used SAF and number subbands, improving the system performance BAF methods. The proposed technique compared to and attaining lower complexity, whereas the MT method the other methods for small values of M has higher of ournal fails to converge and the performance of the DFT-MDF computational complexity. The new technique works method deteriorates. very well with a larger number of subbands, improving obal J the system performance and attaining lower complexity, Gl whereas the MT method fails to converge and the performance of the DFT-MDF method deteriorates. XII. Future Scope Adaptive digital signal processing is a rapidly growing branch of DSP and has great significance in the design of adaptive systems. The various signal processing applications demand for reduction in trade off between misadjustment and convergence rate,

taking realization of algorithm into account. The 10. P. Vaidyanathan, Multirate Systems and Filter modifications resulted in a new subband method that, Banks. Englewood Cliffs, NJ:Prentice -Hall, 1993 unlike existing methods, improves the performance and 11. J. Huo, S. Nordholm, and Z. Zang, “New weight reduces the computational complexity for a large transform schemes for delayless subband adaptive number of subbands. There is a scope of improvement filtering,” in Proc. GLOBECOM’01, New York, Nov. in replacing the existing time domain adaptive filters with 2001, pp. 197–201. frequency domain adaptive filters. There’s a lot, which 12. L. Larson, J. de Haan, and I. Claesson, “A new can be done in future for improvement on the methods subband weight transforms for delayless subband for noise cancellation. The field of digital signal adaptive filtering structures,” in Proc. Digital Signal processing and in particular adaptive filtering is vast and Process. Workshop, Aug. 2002, pp. 201–206. further research and development in this area can result 13. J. M. de Haan, N. Grbic, I. Claesson, and S. E. in some improvement on the methods studied in this Nordholm, “Filter bank design for subband adaptive paper. microphone arrays,” IEEE Trans. Speech Audio

2014 Process., vol. 11, no. 1, pp. 14–23, Jan. 2003. XIII. Acknowledgements 14. A. H. Sayed, Fundamentals of Adaptive Filtering . Year The work is carried out through the research New York: Wiley, 2003. facility at the Department of Computer Science & 15. Z. Lin and Y. Liu, “Design of complex fir filters with 50 Engineering, Department of Electronics & reduced group delay error using semidefinite Communication Engineering and Department of programming,” IEEE Trans. Signal Process., vol. 13,

Electrical & Electronics Engineering, JITS, Nustulapur no. 9, pp 529–532, Sep. 2006. (V), Karimnagar-District, A.P., Thanks to the experts who 16. R. Venkataramani and Y. Bresler, “Filter design for have contributed towards development of this paper. MIMO sampling and reconstruction,” IEEE Trans. Signal Process., vol. 51, no. 12, pp. 3164–3167, References Références Referencias Dec. 2003. 17. Y. C. Lim, J. Lee, C. K. Chen, and R. Yang, “A 1. D. R. Morgan and J. C. Thi, “A delayless subband weighted least squares algorithm for quasi- adaptive filter architecture, ”IEEE Trans. Signal equiripple FIR and IIR digital filter design,” IEEE Process., vol. 43, no. 8, pp. 1819–1830, Aug. 1995.

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um Mar. 1992. pproach to frequency domain and subband 18. H. Gustafsson, S. E. Nordholm, and I. Claesson, Vol adaptive filtering,” IEEE Trans. Signal Process., vol. “Spectral subtraction using reduced delay 48, no. 9, pp. 2607 2619, Sep. 2000. F

() convolution and adaptive averaging,” IEEE Trans. 3. P. Loizou, Speech Enhancement: Theory and Speech Audio Process., vol. 9, no. 8, pp. 799–807, Practice. Boca Raton, FL: CRC, 2007. Nov. 2001. 4. V. R. Ramachandran, G. Kannan, A. A. Milani, and I. 19. A. V. Oppenheim and R. W. Schafer, Discrete-Time M. S. Panahi, “Speech enhancement in functional Signal Processing, 2nd ed. Englewood Cliffs, NJ: MRI environment using adaptive sub-band algorithms, in Proc. ICASSP’07, 2007, pp. II-341–II- Prentice-Hall, 1999.

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